Questions tagged [dual-spaces]

The dual space of a vector space $V$ over a field $k$ is the vector space of all linear maps from $V$ into $k$.

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Exercise 3.F.5 in "Linear Algebra Done Right 3rd Edition" by Sheldon Axler.

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. 3.94 Definition dual space, $V'$ The dual space of $V$, denoted $V'$, is the vector space of all linear functionals on ...
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Is $A\subset X$ bounded when $\{\Delta(x):x\in A\}$ is bounded for all $\Delta\in X^*$, $X$ is a normed space and $A\subset X$? [closed]

Let $X$ a normed space and $A\subset X$. Suppose that for all $\Delta\in X^*$, $\{\Delta(x):x\in A\}$ is bounded. Prove that A is bounded
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Weak convergence and duals for $L_p$ involving time and probability space

Questions are from the theory of PDEs\SPDEs Question 1. Suppose $(V, H, V^\star)$ is a Gelfand triple (embeddings are continuous and dense, so $\|\|_H \le C \|\|_V$ for some $C>0$ etc) of ...
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Proof about dual space

Let $X$ a vectorial space y let $\Gamma \subset X^{\ast}$. We will say that $\Gamma$ is total in $X$ if $f(x)=0$, $\forall f \in \Gamma$ implies that $x=0$. I have to prove that if $\Gamma$ is total ...
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Is there a very small gap or no gap in this proof? ("Linear Algebra Done Right 3rd Edition" by Sheldon Axler.)

I am reading "Linear Algebra Done Right 3rd Edition" by Sheldon Axler. Let $V$ be a vector space. Let $V'$ be the dual space of $V$. Let $W$ be a vector space. Let $W'$ be the dual space of $...
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What's the problem with the evaluation map not being continuous?

When introducing differentiable functions between locally convex spaces, many authors (e.g. Bastiani, Keller, Kriegl-Michor) notice that the evaluation map $$ E\times E^*\to\mathbb R,\qquad (x,L)\...
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Irreducible dual representation on finite dimensional vector space

I started reading Brian C.Hall's book on matrix Lie group and Lie algebra representations and I come across the following: Dual representation is irreducible if and only if the representation is ...
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How to prove if the following is a dual vector?

I have just started learning dual spaces and in my current understanding, I know that dual vectors are vectors that maps vectors onto scalars. However, how do I prove if a vector is a dual vector? For ...
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Proof for $(Z^\perp)^{\perp} = Z$

Let $V$ be a vector space of dimension n over an arbitrary field $F$, and let $V^{*}$ be the dual space of $V$. The definition of annihilator is $Z^{\perp} = \{f \in V^{*} | f(v) = 0 \; \forall v \in ...
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Kernel of a function in the dual space

If I have $f : V^* \rightarrow \mathbb{R}$ in the dual space of $V$, with $dim(V) = n$. I read somewhere that $dim(ker(f)) \ge n-1$, why is this the case? If I take $\{v_1,v_2,...,v_n\}$ to be a basis ...
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Clarification on definition of representation of $\mathrm{Hom}(V,W)$

From Representation Theory by W. Fulton and J. Harris: Let $V$ be a finite dimensional vector space, and $G$ a finite group. Let $\rho: G \to \mathrm{GL}(V)$ be a representation of $V$. The dual of ...
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Let $M\subseteq X$ be a maximal subspace of the normed space $X$. Is there a functional $f\in X^*$ such that $\ker f = M$?

Let $X$ be a normed space and let $M\subseteq X$ be a maximal subspace. Do we need $M$ to be closed in order to claim that there is a functional $f\in X^*$ such that $\ker f=M$? I proceeded as follows:...
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Question on non-degenerate sesquilinear forms

Let $f$ be a non-degenerate sesquilinear form on $E$, is it true that for all linear $\varphi$ in $E$, so $\varphi \in E^*$, there exist $z \in E$ such that $$\varphi(x) = f(x,z)$$ and if so, why? I ...
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understanding why Wasserstein is weak

I am reading Wasserstein GAN paper and in Appendix A, it says Let $\mathcal{X} \subseteq \mathbb{R}^d$ be a compact set (such as $[0, 1]^d$ the space of images). We define Prob($\mathcal{X}$) to be ...
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Transpose vs Dual vs (complex) Conjugate

I believe I already have come to a reasonable understanding, but speaking with those that I know that are smarter than me, has brought me to where I am. So for the sake of completeness, when is the ...
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How to prove the Non-negativity of a distance function.

Let $U$ be a Banach space and $\Theta$ be a convex function. Let $$\partial\Theta(x):=\{\zeta\in U^*: \Theta(x')-\Theta(x)-\langle \zeta, x'-x\rangle\geq 0\ \forall\ x'\in U\}.$$ Here $\langle \zeta, ...
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$H$-comodule structure of $A\otimes_K A$

I'm studying the paper "Hopf Galois Theory: A survey" by Susan Montgomery, and need some help with something that the paper does not define (and that I'm not able to find anywere). Consider ...
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Showing $L^{\infty}(\mathbb R)^*\neq L^1(\mathbb R)$ using the weak star compactness of unit ball in $L^{\infty}(\mathbb R)^*$.

$\lambda_n(f)=\frac{1}{2n}\int_{-n}^{n}f$ defines a dual element of $ L^{\infty}(\mathbb R)$. It is easy to see that $\lambda_n\in L^{\infty}(\mathbb R)^*_1.$ By using the weak-star compactness of ...
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Weak convergence and convergence of functionals in dual space

Let $X$ be a normed space, $(x_n)_{n \in \mathbb{N}} \subset X$ and $(x_n')_{n \in \mathbb{N}} \subset X'$ such that $(x_n)_{n \in \mathbb{N}}$ weakly converges to $x \in X$ and $(x_n')_{n \in \mathbb{...
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Norm of linear functional on product space

Let $\Phi:L^{p}\times L^p\to\mathbb{R}$ be the linear functional given by $$\Phi(h_0,h_1) = \int_If_0h_0+\int_If_1h_1$$ where $f_0,f_1\in L^{q}$, $1/p+1/q=1$. Consider the norm on $L^p\times L^p$ ...
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A Space is Reflexive if its Image under the Canonical Injection is Reflexive?

Consider the following corollary in Brezis: Here is part of the proof of it: It was mentioned that if $J(E) \subseteq E^{**}$ is reflexive, then $E$ is also reflexive, where $J: E \to E^{**}$ be the ...
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Definition of certain Gateaux differentials of the norm in E. R. Lorch: "A Curvature Study of Convex Bodies in Banach Spaces"

In the paper E. R. Lorch: A Curvature Study of Convex Bodies in Banach Spaces from 1953, the following assumptions and definitions are stated in section II (p. 107-108): Let $(B, \| \cdot \|)$ be a ...
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Reflexivity and Separability of $L^{\infty}$ and $L^{1}$

I am currently trying to understand the following proposition: Let $\mu$ be a Radon measure on $\mathbb{R}^n$. Then: (1) $L^{\infty}(\mu)$ is neither reflexive nor separable (2) $L^{1}(\mu)$ is not ...
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Give a example to show that there not necessarily exist uniformly bounded linear funtionals $\{f_j\}$ such that $f_j(x_k)=\delta_{jk}$.

Suppose $E$ is a normed linear space, $\{x_k\}_{k=1}^{\infty}$ is linearly independent and $\|x_k\|=1, k=1,2,\cdots$. Give a example to show that there not necessarily exist uniformly bounded linear ...
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Proof that dual of $L^p$ is $L^q$

Im working through a proof which shows that for $\frac{1}{p}+\frac{1}{q}$ the map $\Phi:L^q(\Omega;\mathbb{K})\to (L^p(\Omega;\mathbb{K}))^*$, $g\to \Phi(g)[f]:=\int_{\Omega}\overline{g}f\space d\mu$ ...
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Give a example to show that there not necessarily exist $\{e_j\}$ such that $f_i(e_j)=\delta_{ij}, i,j\geq 1$.

Suppose $X$ is a normed linear space. $\{f_i\}\subset X^*$ are linearly independent. Give a example to show that there not necessarily exist $\{e_j\}$ such that $f_i(e_j)=\delta_{ij}, i,j\geq 1$. I ...
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Apparent contradiction between weak metrizability results

I will fix $E \doteq \ell^2$, an infinite dimensional space, with separable dual. In here, there was a discussion that the weak topology of $E$ is not metrizable. By a result given in here, the disk $...
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Let V a real vector space of finite dimension, with a positive definite inner product and let A a symmetric operator in this space

I want to know how can I generalize this excersice. Let V a real vector space of finite dimension, with a positive definite inner product and let A a symmetric operator in this space a)We have $g(v,w)...
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Use natural identification of V with its double dual to prove that $U^{\bot \bot}=U$

I meet this question when doing my homework: Given$U\subset V$,prove that $U^{\bot\bot}=U$,Hint: you may use the natural identification of $V$ with its double dual $V^{**}$,i.e,let $v\in V$ acts on $f\...
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Generalization of the Riesz representation theorem

Any finite dimensional vector space V endowed with nondegenerate bilinear form can be canonically identified with its dual space. I wonder if I can find similar identification for infinite dimensional ...
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Exchange coefficient in tensor product

In the highlighted equation I don't understand why we can exchange the position $\langle a_{(2)}\cdot v^{i}, v_{j} \rangle v^{j}$ to the other element in the tensor product? Is this a property of the ...
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Evaluation under tensor products

Could anyone explain to me the highlighted step in this calculation? Can I just exaluate the tenor product component wise? The first step is basically $$(ev_{V}\otimes id_{V})\circ(id_{V}(v)\otimes ...
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Natural isomorphism between a finite-dimensional vector space and its dual

I was reading about Pontryagin's duality and came accross the following quote a finite-dimensional vector space $V$ and its dual vector space $V^*$ are not naturally isomorphic, but the endomorphism ...
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Is this duality result true, i.e., $\|f\| := \sqrt{\|f\|_1^2 + \|f\|_2^2} \implies |x|^2 =\inf_{y\in E} \left [ |x-y |_1^2 + |y|_2^2 \right ]$?

Let $(E, |\cdot|_1)$ be a normed vector space (n.v.s) and $(E', \| \cdot \|_1)$ its dual. Let $|\cdot|_2$ be an equivalent norm of $|\cdot|_1$ on $E$, and $\| \cdot \|_2$ its dual norm on $E'$. In a ...
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Prove that the canonical embedding from $X$ to $X''$ is isometric

Suppose that $X$ is a normed vector space then the dual space $X'$ of $X$ is a Banach space equipped with the dual norm https://en.wikipedia.org/wiki/Dual_norm, then we can consider $X''$, the double ...
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Is it true that $\|f\|^2 = \sup_{x\in E} \left [ 2 \langle f, x \rangle - |x|^2 \right ]$?

Let $(E, |\cdot|)$ be a normed linear space and $(E', \| \cdot \|)$ its dual. Then $$ \|f\| := \sup_{x\in E} \frac{\langle f, x \rangle}{ |x|}, \quad \forall f \in E'. $$ We have $$ \left [ \frac{\...
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How do I find the base?

Let $W\subset(\mathbb{R}^3)^*$ be a subspace formed by the functionals $f:\mathbb{R^3}\rightarrow \mathbb{R}$ such that $Kerf =[(-4,-4,0),(0,2,2)]$. Find a basis of $W$ The idea I had was: Consider a ...
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The dual space $L_\infty(\mathbb{R})$ can't be identified with $L_1(\mathbb{R})$

Let be $L_\infty(\mathbb{R})$, $L_1(\mathbb{R})$, prove that the dual space of $L_\infty(\mathbb{R})$, $[L_\infty(\mathbb{R})]^*$, can't be identified with $L_1(\mathbb{R})$, sp in other words: $$\...
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Construct an equivalent and strictly convex norm on a separable Banach space whose dual norm is also strictly convex

I'm doing Ex 3.27 in Brezis's book of Functional Analysis. Let $(E, | \cdot |)$ be a separable Banach space and $(E', \| \cdot \|)$ its dual space. Let $(a_n)$ be a countable dense subset of the ...
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The property of duality map in case the dual space is strictly convex

Let $(E, | \cdot |)$ be a Banach space and $(E', \| \cdot \|)$ its dual. Assume that $E'$ is strictly convex. Then for each $x\in E$, there is a unique $f_x \in E'$ such that $\|f_x\|=|x|$ and $\...
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the question of the formulation of the Dunford-Schwartz theorem about $l^{\infty}$

According to the book Dunford-Schwartz, Linear Operators I, Theorem IV.8.16 the space $ (L^{\infty} (S, \Sigma, \nu))^*$ is isomorphic the space $ba (S, \Sigma_1, \| . \|)$. Unfortunately, I ran into ...
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Finding basis given dual basis.

Suppose $V$ is a finite dimensional vector space, $V'$ is its dual space, and $\{\phi_1, \dots, \phi_n\}$ is a basis for $V'$. Suppose I have found $\{v_1, \dots, v_n\}$ with $v_i \in V$ and that $\...
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$T'$ injective $\rightarrow$ $T$ surjective (on finite dimensional spaces)

From a recent qualifying exam at my school: Let $V$ and $W$ be finite dimensional vector spaces and $T: V \rightarrow W$ a linear map. Let $T': W' \rightarrow V'$ be the dual map. Prove that $T$ is ...
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How to prove this conclusion in functional analysis?

Question: Let $\{x_i\}_{i\in \mathbf{N}}$ be a sequence in a normed space $\mathscr{X}$ such that for any $f\in \mathscr{X}^\ast$, \begin{equation} \sum_{i=1}^{\infty}|f(x_i)|<\infty. \end{equation}...
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How does one show a Schauder basis is shrinking?

I feel like I must be missing a trick - I'm self studying functional analysis and have come across Schauder bases, and I'm looking at different classifications e.g. shrinking, boundedly complete, ...
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Is the transpose of $T$ also the dual map of $T$?

Suppose I have $T: V \to W$. Then $T$ defines a map $T': W' \to V'$ that some sources (e.g. Axler) call the dual map of $T$. And $T$ defines a map $T^t: W' \to V'$ that some sources (e.g. Hoffman and ...
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Isomorphism of two representations

Let $\rho:G\to GL(V)$ and $\sigma:G\to GL(W)$ be two representations. We define the representation $\tau:G\to GL(\hom _k (V,W))$ by: $$\tau_{g}\varphi=\sigma_{g}\circ\varphi\circ\rho_{g}^{-1}$$ for $...
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The space spanned by the set of coordinate projections is dense in dual space

This answer leads me to below result. Let $E := \ell^p$ with $1 < p < \infty$. Let $\pi_n: E \to \mathbb R, x \mapsto x_n$ be the canonical projection. Clearly, $\pi_n \in E'$. Let $G := \{\...
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Support functional of a convex set is sublinear

I am studying the "Optimization by Vector Space Methods" book by David G. Luenberger and I am struggling with exercise 19 of the Dual Spaces chapter. How can I show that the support of a ...
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Closed subspace of dual space is the whole space if the intersection of kernels is 0

Let $X$ be a Banach space, and $E \subset X^*$ a subspace of the dual $X^*$ that is closed in the weak-* topology. Show that if $\cap_{\lambda \in E} \ker(\lambda) = 0$, then $E = X^*$. The analogous ...
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